Nuclear Binding Energy: Videos & Practice Problems

Nuclear binding energy plays a crucial role in understanding mass-energy conversions, particularly in the context of isotopes. It is defined as the energy released when an isotope is formed, and conversely, the energy required to break it apart. A key concept is that a higher nuclear binding energy indicates a more stable nucleus for any given isotope.

The formula for calculating nuclear binding energy is derived from Einstein's famous equation, expressed as:

\( E = mc^2 \)

In this equation, \( E \) represents the nuclear binding energy, typically measured in joules (J). The mass defect, denoted as \( m \), must be converted to kilograms (kg) to align with the joules unit, as 1 Joule is equivalent to \( \text{kg} \cdot \text{m}^2/\text{s}^2 \). The speed of light, \( c \), is a constant valued at \( 3.00 \times 10^8 \) meters per second.

Understanding the relationship between nuclear binding energy and mass defect is essential; knowing one allows for the calculation of the other. This interconnection emphasizes the importance of both concepts in nuclear physics and energy calculations.

To calculate the nuclear binding energy of Beryllium-10 in Mega electron volts per mole, we start by determining the mass defect of the isotope. Beryllium-10 has an atomic number of 4, indicating it contains 4 protons and, since its atomic mass is 10, it has 6 neutrons (10 - 4 = 6). The total mass of these subatomic particles is calculated by multiplying the number of each type of particle by their respective masses in atomic mass units (AMU).

The masses of the particles are as follows: protons have a mass of approximately 1.007276 AMU, and neutrons have a mass of about 1.008665 AMU. Thus, the predicted mass of Beryllium-10 can be calculated as:

Predicted mass = (4 protons × 1.007276 AMU) + (6 neutrons × 1.008665 AMU) = 10.08324 AMU.

Next, we find the mass defect by subtracting the actual atomic mass of Beryllium-10 (10.0135347 AMU) from the predicted mass:

Mass defect (Δm) = Predicted mass - Atomic mass = 10.08324 AMU - 10.0135347 AMU = 0.0697053 AMU.

To convert the mass defect from AMU to kilograms, we use the conversion factor where 1 AMU is equal to \(1.66 \times 10^{-27}\) kilograms:

Mass defect in kg = \(0.0697053 \, \text{AMU} \times 1.66 \times 10^{-27} \, \text{kg/AMU} = 1.1571 \times 10^{-28} \, \text{kg}.\)

Now, we can calculate the nuclear binding energy using the formula:

Binding energy (E) = Mass defect (Δm) × \(c^2\),

where \(c\) is the speed of light (\(c \approx 3.00 \times 10^8 \, \text{m/s}\)). Thus, we have:

E = \(1.1571 \times 10^{-28} \, \text{kg} \times (3.00 \times 10^8 \, \text{m/s})^2 = 1.041 \times 10^{-11} \, \text{J}.\)

Since the question requires the binding energy in Mega electron volts per mole, we convert joules to Mega electron volts using the conversion factor \(1 \, \text{MeV} = 1.60 \times 10^{-13} \, \text{J}\):

Binding energy in MeV = \(\frac{1.041 \times 10^{-11} \, \text{J}}{1.60 \times 10^{-13} \, \text{J/MeV}} = 65.087 \, \text{MeV}.\)

Therefore, the nuclear binding energy of Beryllium-10 is approximately 65.087 Mega electron volts per mole.